Vortex Theory for Bodtes Moving tn Water 
Let L(t’) denote this free vortex filament. On the arc By t! Jx ¢1 
e; ? 
we find the union of the vortex filaments, (7) shed during the in- 
terval E pt!) ae Thettotal intensity ofthat union rt Jx, to sah e-y ; 
thus varies with Jy t eet Jy tt it is equal to ee dit dy E (Xt") 
-T, (xX, | = dy [ry (5) tara x, | and it is in the direction towards 
Jy to Similarly we have a union of vortex filaments on the arc 
Ix, to Bo; its intensity at Iy 4: is equal to dy P, (x,t') - Ty (Xx, ‘| 
and it is in the direction towards Bo. 
Let us consider a point Mr on dug, . It is one of the points 
Iy 4: It belongs to the trajectory of a fluid point P which was at an 
anterior time t' ata certain point Bo. lence, Mf, being given, 
Bo, and t' may be considered as determined functions of My and 
t. The same is true for any point Mg, on), f, @ndany point M/ on 
oF , and also for any point M whichis at t ona vortex filament 
! 
hia le pa 
We have supposed that the arc Bo. Biest is infinitely close 
to the arc BoB t on port side bag and nevertheless that Vp on the 
arc Bo, Bi, is not tangent to Le This implies a contradiction of 
the same nature as that encountered in the scheme relative to a steady 
flow about a wing of finite thickness. We have seen in the latter case 
that the relative velocity on the wing is tangent to its trailing edge and 
we however assumed that the free vortices leave the wing in a direc- 
tion orthogonal to this edge. In the present case, the distance of the 
arc Bo, Bi. from the arc B,B; 4 is not really null, for the boun- 
dary layer is not infinitely thin. The contradiction seems to be an in- 
eluctable consequence from the assumption that the fluid is almost 
inviscid, 
We now drop the subscript e and consider that the support 
of the vortex filament YQ (t') at time t is the closed contour BoB, ¢ 
Jx,t'Ix,t1 Bo. Thearc BoB t is bound while the three others are 
free. The intensity of this vortex filament previously determined is 
dis dy [T] (X, t') - I, (X,t')]. The quantity dy[T, (x,t') -T, (x,t!')] is 
positive for the free vortex to be shed from port, but its variation 
with t!' may be positive or negative. 
To make easier the drawing, the angles between By Jx 
and the longitudinal plane of symmetry has been considerably magnifi- 
edin Fig. 5.6. Furthermore, one observes that, if the minimum 
value of X onthe keel is necessarily equal to == (for SoS) plays 
the role of the ordinary trailing edge of a wing), its maximum value 
TZ27 
